2D solidify. Is it possible?
The easy way, which requires a bit (just a bit) of manual work:


Install "Edit Tools 2" and "Loop Tools" addons. They're both
available in basic blender 2.79/2.79a, so just find them in addons
and enable
Select your edge, hit W > Edit Tools > Offset Edges (make sure you're in edge editing mode)

Offset twice, for both positive and negative direction

Select both of edges you've just created. Hit W > Loop Tools > Bridge